Abstract:First-order methods have been advocated for solving optimization problems of large scale. Although they are sometimes the most appropriate techniques, we argue that in many applications it is advantageous to employ second-order information as an integral part of the iteration. This is particularly so when parallel computing environments are available. In this talk, we take a broad view of second-order methods, and center our discussion around three applications: convex L1 regularized optimization, inverse covariance estimation, and nonlinear programming. We note that many efficient methods for these problems can be derived using a semi-smooth Newton framework, which allows us to compare their manifold identification and subspace minimization properties. We propose an algorithm employing a novel active-set mechanism that is of interest in machine learning, PDE-constrained optimization, and other applications. We also discuss dynamic sampling techniques, illustrate their practical performance, and provide work complexity bounds. The talk concludes with some observations about the influence that parallel computing has on large scale optimization calculations.

Biographical sketch:Jorge Nocedal is a professor in the Industrial Engineering Department at Northwestern University. He holds a B.S. degree in physics from the National University of Mexico and a Ph.D in applied mathematics from Rice University. Prior to moving to Northwestern, he taught at the Courant Institute of Mathematical Sciences. He is a SIAM Fellow and an ISI Highly Cited Researcher (mathematics category). In 1998 he was appointed Bette and Neison Harris Professor at Northwestern. Jorge was an invited speaker at the 1998 International Congress of Mathematicians in Berlin. His research focuses on the theory, algorithms and applications of nonlinear programming, and he has developed widely used software, including L-BFGS and Knitro. He is currently Editor-in-Chief of the SIAM Journal on Optimization.

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